A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p
Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 59-70.

Let S/R be a finite extension of discrete valuation rings of characteristic p>0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let 𝔻 S/R be the different of S/R. We show that if S/R is totally ramified and its degree n is a power of p, then any element ρ of L with v L (ρ)-v L (𝔻 S/R )-1(modn) generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

Soit S/R une extension finie d’anneaux de valuation discrète de caractéristique p>0, et supposons que l’extension correspondante L/K des corps de fractions soit séparable et H-Galoisienne pour une K-algèbre de Hopf H. Soit 𝔻 S/R la différente de S/R. Nous montrons que si S/R est totalement ramifiée et que son degré n est une puissance de p alors tout élément ρ de L avec v L (ρ)-v L (𝔻 S/R )-1(modn) engendre L comme H-module. Ce critère est le meilleur possible. Ces résultats généralisent à la situation Hopf-Galoisienne un travail récent de G. G. Elder pour les extensions Galoisiennes.

Published online:
DOI: 10.5802/jtnb.750
Classification: 11S15
Keywords: Normal basis, Hopf-Galois extensions, local fields
Nigel P. Byott 1

1 Mathematics Research Institute University of Exeter Harrison Building North Park Road Exeter EX4 4QF, UK
@article{JTNB_2011__23_1_59_0,
     author = {Nigel P. Byott},
     title = {A valuation criterion for normal basis generators of {Hopf-Galois} extensions in characteristic $p$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {59--70},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {1},
     year = {2011},
     doi = {10.5802/jtnb.750},
     zbl = {1278.11103},
     mrnumber = {2780619},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.750/}
}
TY  - JOUR
TI  - A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2011
DA  - 2011///
SP  - 59
EP  - 70
VL  - 23
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.750/
UR  - https://zbmath.org/?q=an%3A1278.11103
UR  - https://www.ams.org/mathscinet-getitem?mr=2780619
UR  - https://doi.org/10.5802/jtnb.750
DO  - 10.5802/jtnb.750
LA  - en
ID  - JTNB_2011__23_1_59_0
ER  - 
%0 Journal Article
%T A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$
%J Journal de Théorie des Nombres de Bordeaux
%D 2011
%P 59-70
%V 23
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.750
%R 10.5802/jtnb.750
%G en
%F JTNB_2011__23_1_59_0
Nigel P. Byott. A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$. Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 59-70. doi : 10.5802/jtnb.750. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.750/

[1] N. P. Byott, Integral Hopf-Galois structures on degree p 2 extensions of p-adic fields. J. Algebra 248 (2002), 334–365. | MR: 1879021 | Zbl: 0992.11065

[2] N. P. Byott and G. G. Elder, A valuation criterion for normal bases in elementary abelian extensions. Bull. London Math. Soc. 39 (2007), 705–708. | MR: 2365217 | Zbl: 1128.11055

[3] L. N. Childs, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory. Mathematical Surveys and Monographs 80, American Mathematical Society, 2000. | MR: 1767499 | Zbl: 0944.11038

[4] G. G. Elder, A valuation criterion for normal basis generators in local fields of characteristic p. Arch. Math. 94 (2010), 43–47. | MR: 2581332 | Zbl: pre05675006

[5] C. Greither and B. Pareigis, Hopf Galois theory for separable field extensions. J. Algebra 106 (1987), 239–258. | MR: 878476 | Zbl: 0615.12026

[6] J.-P. Serre, Local Fields. Graduate Texts in Mathematics 67, Springer, 1979. | MR: 554237 | Zbl: 0423.12016

[7] J.-P. Serre, Linear Representations of Finite Groups. Graduate Texts in Mathematics 42, Springer, 1977. | MR: 450380 | Zbl: 0355.20006

[8] H. Stichtenoth, Algebraic Function Fields and Codes. Springer, 1993. | MR: 1251961 | Zbl: 1155.14022

[9] L. Thomas, A valuation criterion for normal basis generators in equal positive characteristic. J. Algebra 320 (2008), 3811–3820. | MR: 2457723 | Zbl: pre05380154

Cited by Sources: